Euler Paths and Euler Circuits - University of Kansas

Euler Paths and Euler Circ

An Euler path is a path that u
exactly once.
An Euler circuit is a circuit tha
exactly once.

An Euler path starts and en
An Euler circuit starts and

cuits

uses every edge of a graph
at uses every edge of a graph

nds at different vertices.
ends at the same vertex.

Euler Paths and Euler Circ

B
A

C

ED

An Euler path:

cuits

B
A

C

E D

BBADCDEBC

Euler Paths and Euler Circ

B
A

C

ED

Another Euler pat

cuits

B
A

C

E D

th: CDCBBADEB

Euler Paths and Euler Circ

B
A

C

ED

An Euler circuit:

cuits

B
A

C

E D

CDCBBADEBC

Euler Paths and Euler Circ

B
A

C

ED

Another Euler circ

cuits

B
A

C

E D

cuit: CDEBBADC

Euler Paths and Euler Circ

Is it possible to determine w
Euler path or an Euler circui
having to find one explicitly?

cuits

whether a graph has an
it, without necessarily
?

The Criterion for Euler Pa

Suppose that a graph has an Eu

aths

uler path P.

The Criterion for Euler Pa

Suppose that a graph has an Eu

For every vertex v other than th
the path P enters v the same n
(say s times).

aths

uler path P.
he starting and ending vertices,
number of times that it leaves v

The Criterion for Euler Pa

Suppose that a graph has an Eu

For every vertex v other than th
the path P enters v the same n
(say s times).

Therefore, there are 2s edges

aths

uler path P.
he starting and ending vertices,
number of times that it leaves v

having v as an endpoint.

The Criterion for Euler Pa

Suppose that a graph has an Eu

For every vertex v other than th
the path P enters v the same n
(say s times).

Therefore, there are 2s edges

Therefore, all vertices other
P must be even vertices.

aths

uler path P.
he starting and ending vertices,
number of times that it leaves v

having v as an endpoint.
than the two endpoints of

The Criterion for Euler Pa

Suppose the Euler path P starts

aths

s at vertex x and ends at y .

The Criterion for Euler Pa

Suppose the Euler path P starts

Then P leaves x one more time
one fewer time than it enters.

aths

s at vertex x and ends at y .
e than it enters, and leaves y

The Criterion for Euler Pa

Suppose the Euler path P starts
Then P leaves x one more time
one fewer time than it enters.
Therefore, the two endpoints

aths

s at vertex x and ends at y .
e than it enters, and leaves y

of P must be odd vertices.

The Criterion for Euler Pa

The inescapable conclusion (“ba

If a graph G has an Euler
exactly two odd vertices.

Or, to put it another way,

If the number of odd vertic
than 2, then G cannot have

aths

ased on reason alone!”):
path, then it must have

ces in G is anything other
e an Euler path.

The Criterion for Euler Ci

Suppose that a graph G ha

ircuits

as an Euler circuit C .

The Criterion for Euler Ci

Suppose that a graph G ha
For every vertex v in G , ea
endpoint shows up exactly

ircuits

as an Euler circuit C .
ach edge having v as an
once in C .

The Criterion for Euler Ci

Suppose that a graph G ha

For every vertex v in G , ea
endpoint shows up exactly

The circuit C enters v the
leaves v (say s times), so v

ircuits

as an Euler circuit C .

ach edge having v as an
once in C .

same number of times that it
v has degree 2s.

The Criterion for Euler Ci

Suppose that a graph G ha

For every vertex v in G , ea
endpoint shows up exactly

The circuit C enters v the
leaves v (say s times), so v

That is, v must be an ev

ircuits

as an Euler circuit C .

ach edge having v as an
once in C .

same number of times that it
v has degree 2s.

ven vertex.

The Criterion for Euler Ci

The inescapable conclusion (“ba

If a graph G has an Euler circ
must be even vertices.

Or, to put it another way,

If the number of odd vertic
than 0, then G cannot have

ircuits

ased on reason alone”):
cuit, then all of its vertices

ces in G is anything other
e an Euler circuit.

Things You Should Be Wo

Does every graph with zer
circuit?

Does every graph with tw
path?

Is it possible for a graph ha

ondering

ro odd vertices have an Euler
wo odd vertices have an Euler

ave just one odd vertex?

How Many Odd Vertices?

?

How Many Odd Vertices?

Odd ve

?

ertices

How Many Odd Vertices?

68

Number of od

24

?

8 6
8
dd vertices

4

The Handshaking Theorem

The Handshaking Theorem says

In every graph, the sum of
equals twice the number o

If there are n vertices V1, . . . , V
there are e edges, then

d1 + d2 + · · · +

Or, equivalently,
e = d1 + d2 + ·

m

s that
f the degrees of all vertices
of edges.
Vn, with degrees d1, . . . , dn, and

dn−1 + dn = 2e

· · · + dn−1 + dn
2

The Handshaking Theorem

Why “Handshaking”?

If n people shake hands, and th
times, then the total number of

d1 + d2 + · · ·
2

(How come? Each handshake in
number d1 + d2 + · · · + dn−1 +
twice.)

m

he i th person shakes hands di
f handshakes that take place is
+ dn−1 + dn .
2

nvolves two people, so the
dn counts every handshake